proving equivalence of cauchy sequence definitions in real number? There is definition of Cauchy sequence in the book of Introduction to Calculus and classical analysis by Omar hijab and that is :
$\forall n,m\in \Bbb N  \space e_n \ge 0 ,e_n\to 0 \space ,|a_{m+n}- a_n|\lt e_n  $
$e_n$ is  a error sequence for cauchy sequence .and there is a general definition which many books including Rudin use which is :
$\forall\epsilon\gt0 \space\space\exists N\in\Bbb N  \space \space s.t \space \space\forall m,n\ge N \space\space  |a_n-a_m|\lt\epsilon$
I want to prove that these definitions are equivalent.I don't have any idea how to prove it !
 A: 1) => 2) is quite obvious: $(e_n) \rightarrow 0 $ => For any $ \epsilon >0 $,  there is a N such as: $ n \geq N $ => $ e_n \leq \epsilon $
=> $\forall\epsilon\gt0 \space\space\exists N\in\Bbb N  \space \space s.t \space \space \forall m\in\Bbb N ;\forall n\ge N \space\space  |a_{n+m}-a_n|\lt\epsilon $ 
That is the same as: $\forall\epsilon\gt0 \space\space\exists N\in\Bbb N  \space \space s.t \space \space\forall m,n\ge N \space\space  |a_n-a_m|\lt\epsilon$
2) => 1) : Starting with : $\forall\epsilon\gt0 \space\space\exists N\in\Bbb N  \space \space s.t \space \space\forall m,n\ge N \space\space  |a_n-a_m|\lt\epsilon$
Since one can switch the role of n and m, we can assume that m> n . Let's write m=n+p
2) <=> $ \forall\epsilon\gt0 \space\space\exists N\in\Bbb N  \space \space s.t \space \space\forall p\in\Bbb N,n\ge N \space\space  |a_n-a_{n+p}|\lt\epsilon $
Here the last condition allows you to define : $\epsilon_n = sup(|a_n-a_{n+p}|;p\in\Bbb N)$
since this sequence of p is always bounded.
You then have by definition : $ n \leq N : |a_n-a_{n+p}| \leq \epsilon_n $ ; $ n > N : |a_n-a_{n+p}| \leq \epsilon_n \leq \epsilon $
Hence this sequence can be dominated by an other sequence $(\epsilon_n)$ that $\rightarrow 0$, which is your first definition.
